Let M be a smooth manifold and let E and F be two vector bundles on M. Let

${\displaystyle \Gamma ^{\infty }(E),\ {\hbox{and}}\ \Gamma ^{\infty }(F)}$ 

be the spaces of smooth sections of E and F. An operator

${\displaystyle D:\Gamma ^{\infty }(E)\rightarrow \Gamma ^{\infty }(F)}$ 

is a morphism of sheaves which is linear on sections such that the support of D is non-increasing: supp Ds ⊆ supp s for every smooth section s of E. The original Peetre theorem asserts that, for every point p in M, there is a neighborhood U of p and an integer k (depending on U) such that D is a differential operator of order k over U. This means that D factors through a linear mapping i_D from the k-jet of sections of E into the space of smooth sections of F:

${\displaystyle D=i_{D}\circ j^{k}}$ 

where

${\displaystyle j^{k}:\Gamma ^{\infty }E\rightarrow J^{k}E}$ 

is the k-jet operator and

${\displaystyle i_{D}:J^{k}E\rightarrow F}$ 

is a linear mapping of vector bundles.

Proof edit

The problem is invariant under local diffeomorphism, so it is sufficient to prove it when M is an open set in Rⁿ and E and F are trivial bundles. At this point, it relies primarily on two lemmas:

• Lemma 1. If the hypotheses of the theorem are satisfied, then for every x∈M and C > 0, there exists a neighborhood V of x and a positive integer k such that for any y∈V\{x} and for any section s of E whose k-jet vanishes at y (j^ks(y)=0), we have |Ds(y)|<C.
• Lemma 2. The first lemma is sufficient to prove the theorem.

We begin with the proof of Lemma 1.

Suppose the lemma is false. Then there is a sequence x_k tending to x, and a sequence of very disjoint balls B_k around the x_k (meaning that the geodesic distance between any two such balls is non-zero), and sections s_k of E over each B_k such that j^ks_k(x_k)=0 but |Ds_k(x_k)|≥C>0.

Let ρ(x) denote a standard bump function for the unit ball at the origin: a smooth real-valued function which is equal to 1 on B_1/2(0), which vanishes to infinite order on the boundary of the unit ball.

Consider every other section s_2k. At x_2k, these satisfy

j^2ks_2k(x_2k)=0.

Suppose that 2k is given. Then, since these functions are smooth and each satisfy j^2k(s_2k)(x_2k)=0, it is possible to specify a smaller ball B′_δ(x_2k) such that the higher order derivatives obey the following estimate:

${\displaystyle \sum _{|\alpha |\leq k}\ \sup _{y\in B'_{\delta }(x_{2k})}|\nabla ^{\alpha }s_{k}(y)|\leq {\frac {1}{M_{k}}}\left({\frac {\delta }{2}}\right)^{k}}$ 

where

${\displaystyle M_{k}=\sum _{|\alpha |\leq k}\sup |\nabla ^{\alpha }\rho |.}$ 

Now

${\displaystyle \rho _{2k}(y):=\rho \left({\frac {y-x_{2k}}{\delta }}\right)}$ 

is a standard bump function supported in B′_δ(x_2k), and the derivative of the product s_2kρ_2k is bounded in such a way that

${\displaystyle \max _{|\alpha |\leq k}\ \sup _{y\in B'_{\delta }(x_{2k})}|\nabla ^{\alpha }(\rho _{2k}s_{2k})|\leq 2^{-k}.}$ 

As a result, because the following series and all of the partial sums of its derivatives converge uniformly

${\displaystyle q(y)=\sum _{k=1}^{\infty }\rho _{2k}(y)s_{2k}(y),}$ 

q(y) is a smooth function on all of V.

We now observe that since s_2k and ${\displaystyle \rho }$ _2ks_2k are equal in a neighborhood of x_2k,

${\displaystyle \lim _{k\rightarrow \infty }|Dq(x_{2k})|\geq C}$ 

So by continuity |Dq(x)|≥ C>0. On the other hand,

${\displaystyle \lim _{k\rightarrow \infty }Dq(x_{2k+1})=0}$ 

since Dq(x_2k+1)=0 because q is identically zero in B_2k+1 and D is support non-increasing. So Dq(x)=0. This is a contradiction.

We now prove Lemma 2.

First, let us dispense with the constant C from the first lemma. We show that, under the same hypotheses as Lemma 1, |Ds(y)|=0. Pick a y in V\{x} so that j^ks(y)=0 but |Ds(y)|=g>0. Rescale s by a factor of 2C/g. Then if g is non-zero, by the linearity of D, |Ds(y)|=2C>C, which is impossible by Lemma 1. This proves the theorem in the punctured neighborhood V\{x}.

Now, we must continue the differential operator to the central point x in the punctured neighborhood. D is a linear differential operator with smooth coefficients. Furthermore, it sends germs of smooth functions to germs of smooth functions at x as well. Thus the coefficients of D are also smooth at x.

Let M be a compact smooth manifold (possibly with boundary), and E and F be finite dimensional vector bundles on M. Let

${\displaystyle \Gamma ^{\infty }(E)}$ be the collection of smooth sections of E. An operator

${\displaystyle D:\Gamma ^{\infty }(E)\rightarrow \Gamma ^{\infty }(F)}$ 

is a smooth function (of Fréchet manifolds) which is linear on the fibres and respects the base point on M:

${\displaystyle \pi \circ D_{p}=p.}$ 

The Peetre theorem asserts that for each operator D, there exists an integer k such that D is a differential operator of order k. Specifically, we can decompose

${\displaystyle D=i_{D}\circ j^{k}}$ 

where ${\displaystyle i_{D}}$  is a mapping from the jets of sections of E to the bundle F. See also intrinsic differential operators.

Consider the following operator:

${\displaystyle (Lf)(x_{0})=\lim _{r\to 0}{\frac {2d}{r^{2}}}{\frac {1}{|S_{r}|}}\int _{S_{r}}(f(x)-f(x_{0}))dx}$ 

where ${\displaystyle f\in C^{\infty }(\mathbb {R} ^{d})}$  and ${\displaystyle S_{r}}$  is the sphere centered at ${\displaystyle x_{0}}$  with radius ${\displaystyle r}$ . This is in fact the Laplacian. We show will show ${\displaystyle L}$  is a differential operator by Peetre's theorem. The main idea is that since ${\displaystyle Lf(x_{0})}$  is defined only in terms of ${\displaystyle f}$ 's behavior near ${\displaystyle x_{0}}$ , it is local in nature; in particular, if ${\displaystyle f}$  is locally zero, so is ${\displaystyle Lf}$ , and hence the support cannot grow.

The technical proof goes as follows.

Let ${\displaystyle M=\mathbb {R} ^{d}}$  and ${\displaystyle E}$  and ${\displaystyle F}$  be the rank ${\displaystyle 1}$  trivial bundles.

Then ${\displaystyle \Gamma ^{\infty }(E)}$  and ${\displaystyle \Gamma ^{\infty }(F)}$  are simply the space ${\displaystyle C^{\infty }(\mathbb {R} ^{d})}$  of smooth functions on ${\displaystyle \mathbb {R} ^{d}}$ . As a sheaf, ${\displaystyle {\mathcal {F}}(U)}$  is the set of smooth functions on the open set ${\displaystyle U}$  and restriction is function restriction.

To see ${\displaystyle L}$  is indeed a morphism, we need to check ${\displaystyle (Lu)|V=L(u|V)}$  for open sets ${\displaystyle U}$  and ${\displaystyle V}$  such that ${\displaystyle V\subseteq U}$  and ${\displaystyle u\in C^{\infty }(U)}$ . This is clear because for ${\displaystyle x\in V}$ , both ${\displaystyle [(Lu)|V](x)}$  and ${\displaystyle [L(u|V)](x)}$  are simply ${\displaystyle \lim _{r\to 0}{\frac {2d}{r^{2}}}{\frac {1}{|S_{r}|}}\int _{S_{r}}(u(y)-u(x))dy}$ , as the ${\displaystyle S_{r}}$  eventually sits inside both ${\displaystyle U}$  and ${\displaystyle V}$  anyway.

It is easy to check that ${\displaystyle L}$  is linear:

${\displaystyle L(f+g)=L(f)+L(g)}$  and ${\displaystyle L(af)=aL(f)}$ 

Finally, we check that ${\displaystyle L}$  is local in the sense that ${\displaystyle suppLf\subseteq suppf}$ . If ${\displaystyle x_{0}\notin supp(f)}$ , then ${\displaystyle \exists r>0}$  such that ${\displaystyle f=0}$  in the ball of radius ${\displaystyle r}$  centered at ${\displaystyle x_{0}}$ . Thus, for ${\displaystyle x\in B(x_{0},r)}$ ,

${\displaystyle \int _{S_{r'}}(f(y)-f(x))dy=0}$ 

for ${\displaystyle r'<r-|x-x_{0}|}$ , and hence ${\displaystyle (Lf)(x)=0}$ . Therefore, ${\displaystyle x_{0}\notin suppLf}$ .

So by Peetre's theorem, ${\displaystyle L}$  is a differential operator.

• Peetre, J., Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 7 (1959), 211-218.
• Peetre, J., Rectification à l'article Une caractérisation abstraite des opérateurs différentiels, Math. Scand. 8 (1960), 116-120.
• Terng, C.L., Natural vector bundles and natural differential operators, Am. J. Math. 100 (1978), 775-828.